7 PRIMITIVES AND INTEGRALS 167

g does not have in general a derivative everywhere, and when the deriv-
ative exists it may fail to be equal to/(at the points of a demimerable set):

(8.7.4) (" Change of variables ") Let (p be a real-valued primitive of a
regulated function defined in an interval I; let f be any regulated function
defined in an interval J => <p(l); then, if either f is continuous or (p is monotone,
for any two points a, /? of I, we have

p3 />(/*

/(?«))?'«) <« =

Ja J<P(a)

The only point to check is that /(<p(£))<p'(0 is a regulated function, which
follows at once from the assumptions and from the definition of a regulated
function (Section 7.6); then, if g is a primitive of/, both sides of the formula
are equal to g(q>(fi)) - #(<p(a)), due to (8.4.1).

(8.7.5) (" Integration by parts ") Let f., g be primitives of regulated func-
tions defined in an interval I, and taking their values in two Banach spaces
E, F respectively; and let (x, y) -» [x - y] be a continuous bilinear mapping
0/E x F into a Banach space G; then, for any points a, /? of I

[

Ja

[/CO

Again, the only point to check is that [/•#'] and [f - g] are regulated
functions, and then the formula follows from (8.1.4) and (8.4.1).

8.7.6) Let f be a regulated mapping of I into a Banach space F, and let u
be any continuous linear mapping off into a Banach space G. Then

f *

J a

\
-

/

This follows from (8.4.1) and (8.1.3).

The translation in terms of integrals of the mean value theorem reads:

(8.7.7) For any regulated function f in a compact interval,
fV«) « <

Ja Jrimitive of a regulated function/ the difference g(f$) - g(a),